Transcript PowerPoint-presentasjon

Time Dependent Confounding
and
Marginal Structural Models
Hein Stigum
http://folk.uio.no/heins/
courses
Motivating example
Population:
Exposure:
Mediator:
Outcome:
Alcohol use at two time points
HDL cholesterol
Coronary Heart Disease
HDL
A1
A2
CHD
Time Dependent Confounder:
a confounder (HDL)
that depends on
earlier exposure (A1)
Estimate: Joint effect of alcohol use
Simplified DAG, several variables and arrows missing
Could have HDL1 and HDL2
Common situation in patient-doctor follow up!
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Agenda
• Example
• Concepts
• Time Dependent Confounding
Exercises
– Definition
– Effect measure
– Problem
• Analysis
– Inverse Probability Treatment Weighting
– Marginal Structural Models
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Concepts
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Conditional vs Marginal
Probabilities:
Structural=Causal
Cofactor
Disease
-
+
10%
20%
conditional
15%
marginal
Models:
Ordinary methods remove
confounding by conditioning
C
E
D
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Marginal methods remove
confounding without conditioning
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C
E
D
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Parametric vs Non-parametric
E1 and E2 binary, 4 combinations, want E(D)
A non-parametric analysis allows free estimation:
E(D) for all 4 combinations
A model imposes restrictions:
𝐸 𝐷 = 𝛼0 + 𝛼1 𝑒1 + 𝛼2 𝑒2
A saturated model imposes no restrictions:
𝐸 𝐷 = 𝛼0 + 𝛼1 𝑒1 + 𝛼2 𝑒2 + 𝛼3 𝑒1 𝑒2
A saturated model of binary variables = non-parametric analysis
Will use models in this presentation
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Counterfactual causal effect
• Two possible outcome variables
– Outcome if treated:
– Outcome if untreated:
D1
D0
Counterfactuals
Potential outcomes
• Causal effect
– Individual:
– Average:
D1i-D0i
E(D1-D0)
or other effect measures
Fundamental problem: either D1 or D0 is missing
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Hernan 2004
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Association vs. causation
unexposed
exposed
Causation
Association
vs.
vs.
P(D|E=0)
P(D0)
P(D|E=1)
P(D1)
P(D|E=1)  P(D1)
conditional  marginal
Hernan 2004
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Time dependent confounding
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Time dependence
• Individuals followed over time
• Time varying exposure: E1, E2, …
• Time varying covariates: C1, C2,…
• Outcome:
D
• Focus
– Joint effect of E1, E2, …
– Censoring
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Daniel, Cousens et al. 2013. Rothman, Greenland et al. 2008
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Time dependent confounding
• “Normal” confounding (point exposure)
C
E
D
C is a common cause of E and D
• Time dependent confounding, T1 and T2
C1
C2
E1
E2
D
C is a common cause of E and D
C is affected by E
Conditioning on C will remove confounding
but will cause other problems!
Could also have time-fixed (baseline) confounders
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Exercise: Joint effect
Joint effect:
Assuming that the joint effect
is the sum of the E1 and E2 effect,
what casual paths should be
included in the joint effect?
Causal paths from E1:
DAG
C1
E1
C2
E2
D
C2
E1
E2
D
Causal paths from E2:
5 min
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C1
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C1
C2
E1
E2
D
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Example, anti-HIV treatment
E is anti-HIV treatment
C is CD4 count
D is AIDS
CD41
CD42
anti-HIV1
anti-HIV2
side effect
immune status
AIDS
CD4 count is a time dependent confounder
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Exercise: TimeDependentConfounding, Variants
Variant 1:
C2
E1
E2
D
Variant 1:
a) Show the paths from E1 to D
b) Show the paths from E2 to D
c) Can you estimate the joint effect (E1+E2)
in one ordinary regression model?
Variant 2:
E1
C2
U
E2
D
May have both combined
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Variant 2:
If time, do the same for variant 2
10 min
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Time Dependent Confounder, Summary
• Definition
– A confounder that depends on earlier exposure
• Problems
– Ordinary (conditional) methods do not work
•
•
•
•
•
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Adjustment removes confounding,
but closes a causal path (variant 1),
or opens a collider path (variant 2),
or both
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Examples of
time dependent confounding
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Alcohol and CHD
DAG
Process graph
HDL
HDL
A1
A2
CHD
Alcohol
CHD
The process graph is simpler but less specific
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Patient-Doctor
prognostic
factor
treatment
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Treatment regulated by
level of prognostic factor.
Both affect later disease.
disease
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BMI, exercise and diabetes
BMI
Symmetry:
Either BMI or exercise
could be the exposure
diabetes
exercise
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CO2, temperature and lichen survival
Lichen=“lav”
CO2
temp
lichen
survival
Norwegian animals are democratic,
they eat both “høy” and “lav”
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Statins, cholesterol and CHD
cholesterol
statin
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U
Variant 1 and 2 combined
U=unmeasured common factor,
lifestyle: diet, exercise
CHD
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Examples summary
• Time dependent confounding
– is not rare
– depends on the data generating process
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Four methods,
focus on just one: MSM using IPTW
Analysis under Time Dependent Confounding
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Inverse Probability of Treatment Weighting
C
Simple point treatment (exposure)
E
C
1
0
D
Focus on probability of being exposed (binary)
Subjects
Probabilities
Weights
E
E
E
1
0
300
200
100
400
sum
400
600
1000
"Subjects"
N*w
C
1
0
E
N*w
1
0
400
600
1001
400
600
1000
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sum
800
1200
2000
𝑃(𝐸)
1
C
1
0
0.75
0.33
𝑃(𝐸)
sum
0
0.25 1
0.67 1
1/𝑃(𝐸) 1/𝑃(𝐸)
C
1
0
1
0
1.3
3.0
4.0
1.5
Propensity
scores


C
E
D
Weighted analysis!
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Inverse Probability of Treatment Weighting, Exercise
Sample:
200 females,
800 males,
Sex
100 use Paracet
200 use Paracet
Paracet
D
1. Calculate the risk of Paracet use for each sex.
2. Calculate the RR of Paracet use for females versus males
3. Do an inverse probability of treatment weighting for Paracet.
4. Calculate the RR of Paracet use for females versus males in the
reweighted pseudo data
5. Explain the results in the DAG
8 min
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Inverse Probability of Treatment Weighting
Time varying treatment (exposure)
Weights:
C1
C2
E1
E2
D
w1
w2
Weight at E2:
Weight for the entire exposure and covariate history
up to time 2
E is treatment, D is disease
C is a prognostic factor
Weight by w1*w2
Ordinary
weights:
Stabilized
weights:
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𝑡
𝑤 𝑡 =
𝑘=1
𝑡
𝑤𝑠 𝑡 =
𝑘=1
Time points t1 and t2
1
𝑃(exposure at k|cov. and exp. up to k)
𝑃(exposure at k|exposure up to k)
𝑃(exposure at k|cov. and exp. up to k)
Vary a lot
Vary less
Cole and Hernan 2008. Veieroed, Lydersen et al. 2012. Daniel, Cousens et al. 2013
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Marginal Structural Model
C1
E1
C2
ws
E2
DAG for the reweighted pseudo data
D
MSM:
The expected value of a counterfactual outcome D
under a hypothetical exposure 𝑒=(e1,e2):
𝐸 𝐷𝑒 = 𝛼0 + 𝛼1 𝑒1 + 𝛼2 𝑒2
Joint effect = 𝐸 𝐷1 − 𝐸 𝐷0
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Veieroed, Lydersen et al. 2012. Daniel, Cousens et al. 2013. Rothman, Greenland et al. 2008
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MSM in Stata
* Probability of E1
regress E1 C1
C1
C2
E1
E2
predict p1
replace p1=1-p1 if E1==0
D
* Probability of E2
regress E2 E1 C1 C2
predict p2
replace p2=1-p2 if E2==0
* Weights
generate w=1/(p1*p2)
𝑡
𝑤 𝑡 =
𝑘=1
* MSM
1
𝑃(exposure at k|cov. and exp. up to k)
𝐸 𝐷𝑒 = 𝛼0 + 𝛼1 𝑒1 + 𝛼2 𝑒2
regress D E1 E2 [pw=w]
Easy-peasy!
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Alcohol, HDL and CHD
Example
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Generalized Linear Models
• Model: Distribution and Link
• Distribution family
– Given by data
Binomial
• Link function
– May chose
Identity
Linear binomial model
RD
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H.S.
McCullagh and Nelder 1989
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Example: Alcohol, HDL and CHD
All variables binary (0/1), 1:
Exposure: Alcohol use at two time points
Mediator: HDL cholesterol
Outcome: Coronary Heart Disease
high HDL
high 85 %
Alcohol
low
75 %
RD
10 %
high alcohol
high HDL
disease
HDL
+0.1
A1
HDL
-0.3
A2
-0.2
-0.05
high
low
RD
CDH
1%
21 %
-20 %
CHD
-0.02
time
the year before
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last year
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now
Ruidavets, Ducimetiere et al. 2002
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Ex cont: Estimated effects
HDL
+0.1
A1
-0.3
A2
-0.2
-0.05
CHD
-0.02
Conditional
models
Estimated effects
A1:
-0.02+0.1*(-0.2)=
A2:
-0.05=
MSM: E(D1,1)-E(D0,0)=
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Marginal
models
True Crude Adjusted IPTW
-0.04 -0.04 -0.02
-0.04
-0.05 0.01
-0.05
-0.05
-0.09
-0.09
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IPTWs
-0.04
-0.05
-0.09
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Only T2 measures
• Structure almost as before, but T1 not measured
C
HDL
+0.1
A1
+0.5
-0.3
A2
-0.2
-0.05
𝑏𝑖𝑎𝑠 𝑏𝐸𝐷 =
CHD
U
Path
1 A2→CHD
Type
causal
Status
open
OK
2 A2←[HDL]→CHD
non-cau closed
OK
3 A2←A1→CHD
non-cau open
?
4 A1→CHD
causal
-
-
5 A1→HDL→CHD
causal
-
-
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bCD
E
𝐶𝑟𝑢𝑑𝑒 − 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 = 𝑏𝐶𝐸 𝑏𝐶𝐷
-0.02
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bCE
Confounding:
D
𝑣𝑎𝑟(𝐶)
𝑣𝑎𝑟(𝐸)
Estimate
-0.05
-0.07 bias in the right direction
-0.04 not included
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Summing up
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Time Dependent Confounder
• Definition
– a confounder that depends on earlier exposure
• Presence
– is not rare
– depends on the data generating process
• Methods
–
–
–
–
–
Standard (conditional) fail
Marginal Structural Models with Inverse Probability Treatment Weighting
G-computation
G-estimation of Structural Nested Models
new
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M
arginal
S
tructural
M
odels
• Pro
– Weighting removes confounding
– can also handle informative censoring
– Standard software (not SPSS)
– Outcomes: continuous, binary, count,
time-to-event…
• Con
– IPTW can be inefficient, stabilization helps
– Exposures: continuous are difficult
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Recommended reading
• DAGs
– Veieroed M, Lydersen S, Laake P. 2012. Medical statistics in
clinical and epidemiological research.
• Time Dependent Confounding
– Daniel, R. M., S. N. Cousens, B. L. De Stavola, M. G.
Kenward and J. A. C. Sterne (2013). "Methods for dealing
with time-dependent confounding." Statistics in Medicine
32(9): 1584-1618.
– Rothman, K. J., S. Greenland and T. L. Lash (2008). Modern
Epidemiology.
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References
•
Cole, S. R. and M. A. Hernan (2008). "Constructing inverse probability weights for marginal
structural models." Am J Epidemiol 168(6): 656-664.
•
Daniel, R. M., S. N. Cousens, B. L. De Stavola, M. G. Kenward and J. A. C. Sterne (2013).
"Methods for dealing with time-dependent confounding." Statistics in Medicine 32(9): 15841618.
•
Hernan, M. A. (2004). "A definition of causal effect for epidemiological research." J
Epidemiol Community Health 58(4): 265-271.
•
McCullagh, P. and J. Nelder (1989). Generalized Linear Models, Second Edition, Boca
Raton: Chapman and Hall.
•
Petersen, M. L., S. E. Sinisi and M. J. van der Laan (2006). "Estimation of direct causal
effects." Epidemiology 17(3): 276-284.
•
Rothman, K. J., S. Greenland and T. L. Lash (2008). Modern Epidemiology. Philadelphia,
Lippincott Williams & Wilkins.
•
Ruidavets, J. B., P. Ducimetiere, D. Arveiler, P. Amouyel, A. Bingham, A. Wagner, D. Cottel,
B. Perret and J. Ferrieres (2002). "Types of alcoholic beverages and blood lipids in a French
population." J Epidemiol Community Health 56(1): 24-28.
•
Ten Have, T. R. and M. M. Joffe (2012). "A review of causal estimation of effects in
mediation analyses." Statistical Methods in Medical Research 21(1): 77-107.
•
Veieroed, M., S. Lydersen and P. Laake (2012). Medical statistics in clinical and
epidemiological research. Oslo, Gyldendal Akademisk.
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Extra material
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Process graphs
• Notation
– Variables over time
– replaced by process
P1
P2
P3
P
– One process may drive another
P
S
– Feedback loops
P
S
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DAGs and process graphs
DAG
X1
Y1
Z1
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X2
Y2
Z2
Process
X3
Y3
Z3
…
…
…
X
Y
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Z
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TimeDependentConfounding as process
DAG
C1
Process
C
C2
E
E1
E2
D
D
Conditions for TimeDependentConfounding
1) C is a confounder for
2) C is a mediator for
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E on D
E on D
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IPTW for time varying exposures
Courtesy of JM Gran
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Courtesy of JM Gran
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Matching vs. InverseProbabilityTreatmentWeighting
unexposed
Matching
IPTW
vs.
vs.
unexposed
matches
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exposed
P(D0)
exposed
HS
P(D1)
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History of counterfactual modeling
• Goes back to Neyman (1923), Fisher
(1935) and Cochran and Cox (1950)
• Formalized by Rubin (1974 and later) typically referred to as the potential
outcome framework
• Roots in economic literature through Roy
(1951), Quandt (1972) and Heckman
(1974 and later)
• Extended by Robins (1986 and later)
Courtesy of JM Gran
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Controlled vs. Natural direct effects
Problem:
Exposure, Disease and Mediator
M
E
Want direct effect of E→D,
D
How to treat M?
fix
vary as if no treatment
Controlled direct effect:
exercise
mBMI
Effect of mother’s BMI on Birth Weight
when all exercise the same (fixed)
BW
Natural direct effect:
cholesterol
Effect of statin on CHD
when cholesterol varies as in the absence of treatment
statin
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CHD
Petersen, Sinisi et al. 2006, Ten Have and Joffe 2012
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